Choosing the right probability space for a sample space that contains functions If I want to make some probability measurements for a sample space that would be, for example, all the continuous functions on the interval $ [0,1] $. Is there a "natural" sigma algebra and probability measure to choose for this specific sample space?
I'll give an example to what I call "natural"
for a finite set sample space, a natural sigma algebra would be the power set, the full sigma algebra, and I guess the natural probability measure would be the uniform measure.
Im intresed in question such "what is the probability of a continuois function to.be differentiable" and Im looking for the suitable probability soace that would make the question well defined.
Thanks in advance.
 A: The space $C([0,1])$ of continuous functions on $[0,1]$ can be equipped with a natural norm, which is the supremum norm (inducing the topology of uniform convergence). That makes $C([0,1])$ into a Banach space (and in fact also a Polish space). Hence I would say that the most "natural" $\sigma$-algebra on $C([0,1])$ is its Borel $\sigma$-algebra $\mathcal{B}_{C([0,1])}$, generated by all the open sets.
However, in the example you gave with differentiability it is not clear whether the set of differentiable function would be a Borel set. (I actually suspect it might not be; similar to the fact that the set of continuous functions is not measurable with respect to the product $\sigma$-algebra $\mathcal{B}_{\mathbb{R}}^{\otimes [0,1]}$ on $\mathbb{R}^{[0,1]}$).
I am also not sure what you would consider to be a "natural" probability measure on $C([0,1])$, at least there probably can't be such a thing as a "uniform" distribution  -- I mean, even if you look at $\mathbb{R}$, is not possible as there is no "uniform distribution" on $\mathbb{R}$...
Edit: I found this interesting answer, which deals with the question about the probability of a function being differentiable anywhere
